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Annals of Mathematics Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics By Chongsheng Cao and Edriss S Titi Annals of Mathematics, 166 (2007), 245–267 Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics By Chongsheng Cao and Edriss S Titi Abstract In this paper we prove the global existence and uniqueness (regularity) of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics Introduction Large scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the Navier-Stokes equations, with rotation, coupled to thermodynamics and salinity diffusion-transport equations, which account for the buoyancy forces and stratification effects under the Boussinesq approximation Moreover, and due to the shallowness of the oceans and the atmosphere, i.e., the depth of the fluid layer is very small in comparison to the radius of the earth, the vertical large scale motion in the oceans and the atmosphere is much smaller than the horizontal one, which in turn leads to modeling the vertical motion by the hydrostatic balance As a result one obtains the system (1)–(4), which is known as the primitive equations for ocean and atmosphere dynamics (see, e.g., [20], [21], [22], [23], [24], [33] and references therein) We observe that in the case of ocean dynamics one has to add the diffusion-transport equation of the salinity to the system (1)–(4) We omitted it here in order to simplify our mathematical presentation However, we emphasize that our results are equally valid when the salinity effects are taking into account Note that the horizontal motion can be further approximated by the geostrophic balance when the Rossby number (the ratio of the horizontal acceleration to the Coriolis force) is very small By taking advantage of these assumptions and other geophysical considerations, we have developed and used several intermediate models in numerical studies of weather prediction and long-time climate dynamics (see, e.g., [4], [7], [8], [22], [23], [25], [28], [29], [30], [31] and references therein) Some of these models have also been the subject of analytical mathematical study (see, e.g., [2], [3], [5], [6], [9], [11], [12], [13], [15], [16], [17], [26], [27], [33], [34] and references therein) 246 CHONGSHENG CAO AND EDRISS S TITI In this paper we will focus on the 3D primitive equations in a cylindrical domain Ω = M × (−h, 0), where M is a smooth bounded domain in R2 : (1) (2) (3) (4) ∂v ∂v + (v · ∇)v + w + ∇p + f k × v + L1 v = 0, ∂t ∂z ∂z p + T = 0, ∇ · v + ∂z w = 0, ∂T ∂T + v · ∇T + w + L2 T = Q, ∂t ∂z where the horizontal velocity field v = (v1 , v2 ), the three-dimensional velocity field (v1 , v2 , w), the temperature T and the pressure p are the unknowns f = f0 (β + y) is the Coriolis parameter and Q is a given heat source The viscosity and the heat diffusion operators L1 and L2 are given by 1 Δ− Re1 Re2 1 L2 = − Δ− Rt1 Rt2 L1 = − (5) (6) ∂2 , ∂z ∂2 , ∂z where Re1 , Re2 are positive constants representing the horizontal and vertical Reynolds numbers, respectively, and Rt1 , Rt2 are positive constants which stand for the horizontal and vertical heat diffusion, respectively We set 2 = (∂x , ∂y ) to be the horizontal gradient operator and Δ = ∂x + ∂y to be the horizontal Laplacian We observe that the above system is similar to the 3D Boussinesq system with the equation of vertical motion approximated by the hydrostatic balance We partition the boundary of Ω into: (7) Γu = {(x, y, z) ∈ Ω : z = 0}, (8) Γb = {(x, y, z) ∈ Ω : z = −h}, (9) Γs = {(x, y, z) ∈ Ω : (x, y) ∈ ∂M, −h ≤ z ≤ 0} We equip the system (1)–(4) with the following boundary conditions: winddriven on the top surface and free-slip and non-heat flux on the side walls and bottom (see, e.g., [20], [21], [22], [24], [25], [28], [29], [30]): (10) (11) (12) ∂v ∂T = h τ, w = 0, = −α(T − T ∗ ); ∂z ∂z ∂v ∂T on Γb : = 0, w = 0, = 0; ∂z ∂z ∂v ∂T × n = 0, = 0, on Γs : v · n = 0, ∂n ∂n on Γu : PRIMITIVE EQUATIONS 247 where τ (x, y) is the wind stress on the ocean surface, n is the normal vector to Γs , and T ∗ (x, y) is typical temperature distribution of the top surface of the ocean For simplicity we assume here that τ and T ∗ are time independent However, the results presented here are equally valid when these quantities are time dependent and satisfy certain bounds in space and time Due to the boundary conditions (10)–(12), it is natural to assume that τ and T ∗ satisfy the compatibility boundary conditions: ∂τ τ · n = 0, (13) × n = 0, on ∂M ∂n ∂T ∗ (14) =0 on ∂M ∂n In addition, we supply the system with the initial condition: (15) v(x, y, z, 0) = v0 (x, y, z) (16) T (x, y, z, 0) = T0 (x, y, z) In [20], [21] and [33] the authors set up the mathematical framework to study the viscous primitive equations for the atmosphere and ocean circulation Moreover, similar to the 3D Navier-Stokes equations, they have shown the global existence of weak solutions, but the question of their uniqueness is still open The short time existence and uniqueness of strong solutions to the viscous primitive equations model was established in [15] and [33] In [16] the authors proved the global existence and uniqueness of strong solutions to the viscous primitive equations in thin domains for a large set of initial data whose size depends inversely on the thickness of the domain In this paper we show the global existence, uniqueness and continuous dependence on initial data, i.e global regularity and well-posedness, of the strong solutions to the 3D viscous primitive equations model (1)–(16) in a general cylindrical domain, Ω, and for any initial data It is worth stressing that the ideas developed in this paper can equally apply to the primitive equations subject to other kinds of boundary conditions As in the case of 3D Navier-Stokes equations the question of uniqueness of the weak solutions to this model is still open Preliminaries 2.1 New Formulation First, let us reformulate the system (1)–(16) (see also [20], [21] and [33]) We integrate the equation (3) in the z direction to obtain w(x, y, z, t) = w(x, y, −h, t) − z −h ∇ · v(x, y, ξ, t)dξ By virtue of (10) and (11) we have (17) w(x, y, z, t) = − z −h ∇ · v(x, y, ξ, t)dξ, 248 CHONGSHENG CAO AND EDRISS S TITI and (18) −h ∇ · v(x, y, ξ, t)dξ = ∇ · −h v(x, y, ξ, t)dξ = We denote (19) φ(x, y) = h −h φ(x, y, ξ)dξ, ∀ (x, y) ∈ M In particular, (20) v(x, y) = h −h v(x, y, ξ)dξ, in M, denotes the barotropic mode We will denote by v = v − v, (21) the baroclinic mode, that is the fluctuation about the barotropic mode Notice that (22) v = Based on the above and (12) we obtain ∇ · v = 0, (23) in M, and ∂v × n = 0, ∂n By integrating equation (2) we obtain (24) v · n = 0, p(x, y, z, t) = − on ∂M z −h T (x, y, ξ, t)dξ + ps (x, y, t) Substituting (17) and the above relation into equation (1), we reach (25) ∂v + (v · ∇)v − ∂t z −h + ∇ps (x, y, t) − ∇ ∇ · v(x, y, ξ, t)dξ z −h ∂v ∂z T (x, y, ξ, t)dξ + f k × v + L1 v = Remark Notice that due to the compatibility boundary conditions (13) and (14) one can convert the boundary condition (10)–(12) to be homoge2 neous by replacing (v, T ) by (v + (z+h) 2−h /3 τ, T + T ∗ ) while (23) is still true For simplicity and without loss of generality we will assume that τ = 0, T ∗ = However, we emphasize that our results are still valid for general τ and T ∗ provided they are smooth enough In a forthcoming paper we will study the long-time dynamics and global attractors to the primitive equations with general τ and T ∗ 249 PRIMITIVE EQUATIONS Therefore, under the assumption that τ = 0, T ∗ = 0, we have the following new formulation for system (1)–(16): (26) z ∂v + L1 v + (v · ∇)v − ∂t + ∇ps (x, y, t) − ∇ (27) ∂v ∂z −h = 0, z=0 ∂v ∂z z=−h −h ∂v ∂z T (x, y, ξ, t)dξ + f k × v = 0, z ∂T + L2 T + v · ∇T − ∂t (28) z ∇ · v(x, y, ξ, t)dξ −h ∇ · v(x, y, ξ, t)dξ = 0, v · n|Γs = 0, ∂T = Q, ∂z ∂v ×n ∂n = 0, Γs (∂z T + αT )|z=0 = 0; ∂z T |z=−h = 0; ∂n T |Γs = 0, (29) (30) v(x, y, z, 0) = v0 (x, y, z), (31) T (x, y, z, 0) = T0 (x, y, z) 2.2 Properties of v and v By taking the average of equations (26) in the z direction, over the interval (−h, 0), and using the boundary conditions (28), we obtain the following equation for the barotropic mode (32) ∂v + (v · ∇)v − ∂t h −∇ −h z −h z −h ∇ · v(x, y, ξ, t)dξ ∂v + ∇ps (x, y, t) ∂z T (x, y, ξ, t)dξdz + f k × v − Δv = Re1 As a result of (22), (23) and integration by parts, (33) (v · ∇)v − z −h ∇ · v(x, y, ξ, t)dξ ∂v = (v · ∇)v + [(v · ∇)v + (∇ · v) v] ∂z By subtracting (32) from (26) and using (33) we obtain the following equation for the baroclinic mode (34) ∂v + L1 v + (v · ∇)v − ∂t z −h ∇ · v(x, y, ξ, t)dξ ∂v ∂z + (v · ∇)v + (v · ∇)v − [(v · ∇)v + (∇ · v) v] z z T (x, y, ξ, t)dξ − T (x, y, ξ, t)dξdz −∇ h −h −h −h + f k × v = Therefore, v satisfies the following equations and boundary conditions: 250 CHONGSHENG CAO AND EDRISS S TITI ∂v Δv + (v · ∇)v + [(v · ∇)v + (∇ · v) v] + f k × v − ∂t Re1 z + ∇ ps (x, y, t) − T (x, y, ξ, t) dξ dz = 0, h −h −h (35) ∇ · v = 0, in M, ∂v v · n = 0, (37) × n = 0, on ∂M, ∂n and v satisfies the following equations and boundary conditions: (36) (38) z ∂v + L1 v + (v · ∇)v − ∂t −h ∇ · v(x, y, ξ, t)dξ ∂v ∂z + (v · ∇)v + (v · ∇)v − [(v · ∇)v + (∇ · v) v] + f k × v z z T (x, y, ξ, t)dξ − T (x, y, ξ, t)dξdz = 0, −∇ h −h −h −h ∂v ∂z (39) = 0, z=0 ∂v ∂z z=−h = 0, v · n|Γs = 0, ∂v ×n ∂n = Γs Remark We recall that by virtue of the maximum principle one is able to show the global well-posedness of the 3D viscous Burgers equations (see, for instance, [19] and references therein) Such an argument, however, is not valid for the 3D Navier-Stokes equations because of the pressure term Remarkably, the pressure term is absent from equation (38) This fact allows us to obtain a bound for the L6 norm of v, which is a key estimate in our proof of the global regularity for the system (1)–(16) 2.3 Functional spaces and inequalities Denote by L2 (Ω), L2 (M ) and H m (Ω), H m (M ) the usual L2 -Lebesgue and Sobolev spaces, respectively ([1]) Let (40) φ p = p Ω |φ(x, y, z)| dxdydz p p , M |φ(x, y)| dxdy p for every φ ∈ Lp (Ω), , for every φ ∈ Lp (M ) Now, V1 = v ∈ C ∞ (Ω) : ∂v ∂z = 0, z=0 v · n|Γs = 0, V2 = T ∈ C ∞ (Ω) : ∂T ∂z ∂v ∂z ∂v ×n ∂n = 0; z=−h = 0, z=−h = 0, ∇ · v = , Γs ∂T + αT ∂z = 0; z=0 ∂T ∂n =0 Γs We denote by V1 and V2 the closure spaces of V1 in H (Ω), and V2 in H (Ω) under H -topology, respectively 251 PRIMITIVE EQUATIONS Definition Let v0 ∈ V1 and T0 ∈ V2 , and let T be a fixed positive time (v, T ) is called a strong solution of (26)–(31) on the time interval [0, T ] if it satisfies (26) and (27) in a weak sense, and also v ∈ C([0, T ], V1 ) ∩ L2 ([0, T ], H (Ω)), T ∈ C([0, T ], V2 ) ∩ L2 ([0, T ], H (Ω)), dv ∈ L1 ([0, T ], L2 (Ω)), dt dT ∈ L1 ([0, T ], L2 (Ω)) dt For convenience, we recall the following Sobolev and Ladyzhenskaya’s inequalities in R2 (see, e.g., [1], [10], [14], [18]): (41) φ L4 (M ) ≤ C0 φ 1/2 L2 (42) φ L8 (M ) ≤ C0 φ 3/4 L6 (M ) φ 1/2 H (M ) , φ 1/4 H (M ) , for every φ ∈ H (M ), and the following Sobolev and Ladyzhenskaya’s inequalities in R3 (see, e.g., [1], [10], [14], [18]): (43) u L3 (Ω) ≤ C0 u 1/2 L2 (Ω) (44) u L6 (Ω) ≤ C0 u H (Ω) , u 1/2 H (Ω) , for every u ∈ H (Ω) Here C0 is a positive constant which might depend on the shape of M and Ω but not on their size Moreover, by (41) we get φ |φ|3 L4 (M ) ≤ C0 φ (45) L6 (M ) 12 L12 (M ) = ≤ C0 |φ|3 L2 (M ) |φ|3 H (M ) |φ|4 |∇φ|2 dxdy + φ M 12 L6 (M ) , for every φ ∈ Also, we recall the integral version of Minkowsky inp spaces, p ≥ Let Ω ⊂ Rm1 and Ω ⊂ Rm2 be two equality for the L measurable sets, where m1 and m2 are positive integers Suppose that f (ξ, η) is measurable over Ω1 × Ω2 Then, H (M ) 1/p p |f (ξ, η)|dη (46) Ω1 Ω2 dξ 1/p ≤ |f (ξ, η)|p dξ Ω2 dη Ω1 A priori estimates In the previous subsections we have reformulated the system (1)–(16) and obtained the system (26)–(31) The two systems are equivalent when (v, T ) is a strong solution The existence of such a strong solution for a short interval of time, whose length depends on the initial data and the other physical parameters of the system (1)–(16), was established in [15] and [33] Let 252 CHONGSHENG CAO AND EDRISS S TITI (v0 , T0 ) be given initial data In this section we will consider the strong solution that corresponds to the initial data in its maximal interval of existence [0, T∗ ) Specifically, we will establish a priori upper estimates for various norms of this solution in the interval [0, T∗ ) In particular, we will show that if T∗ < ∞ then the H norm of the strong solution is bounded over the interval [0, T∗ ) This key observation plays a major role in the proof of global regularity of strong solutions to the system (1)–(16) 3.1 L2 estimates We take the inner product of equation (27) with T , in and obtain L2 (Ω), 1d T dt 2 + ∇T Rt1 2 + Tz Rt2 QT dxdydz − = 2 + α T (z = 0) z v · ∇T − Ω −h Ω 2 ∂T ∂z ∇ · v(x, y, ξ, t)dξ T dxdydz After integrating by parts we get (47) − z v · ∇T − −h Ω ∂T ∂z ∇ · v(x, y, ξ, t)dξ T dxdydz = As a result of the above we conclude 1d T dt 2 + Tz Rt2 2 QT dxdydz ≤ Q ∇T Rt1 = 2 + + α T (z = 0) T 2 Ω Notice that (48) 2 T ≤ 2h2 Tz 2 + 2h T (z = 0) Using (48) and the Cauchy-Schwarz inequality we obtain d T dt (49) 2 + Tz + α T (z = 0) Rt2 h ≤ 2(h2 Rt2 + ) Q α ∇T Rt1 (50) 2 + 2 By the inequality (48) and thanks to Gronwall inequality the above gives (51) T 2 − ≤e t 2(h2 Rt2 +h/α) T0 2 + (2h2 Rt2 + 2h/α)2 Q , Moreover, we have t (52) ∇T (s) Rt1 2 + Tz (s) Rt2 2 + α T (z = 0)(s) ≤ h2 Rt2 + h α 2 ds Q 2 t + T0 253 PRIMITIVE EQUATIONS By taking the inner product of equation (26) with v, in L2 (Ω), we reach 1d v dt 2 + ∇v Re1 =− 2 + vz Re2 z (v · ∇)v − −h Ω 2 ∇ · v(x, y, ξ, t)dξ z f k × v + ∇ps − ∇ + ∂v · v dxdydz ∂z −h Ω T (x, y, ξ, t)dξ · v dxdydz By integration by parts we get z (v · ∇)v − (53) −h Ω ∂v · v dxdydz = ∂z ∇ · v(x, y, ξ, t)dξ By (36) we have ∇ps · v dxdydz = h (54) ∇ps · v dxdy = −h Ω ps (∇ · v) dxdy = Ω M Since (f k × v) · v = 0, (55) then from (53)–(55) we have 1d v dt 2 + ∇v Re1 2 + vz Re2 z =− −h Ω 2 T (x, y, ξ, t) dξ(∇ · v) dxdydz ≤ h T ∇v By Cauchy-Schwarz and (51) we obtain d v dt 2 + 1 ∇v + vz 2 Re1 Re2 ≤ h2 Re1 T ≤ h2 Re1 T0 2 + (2h2 Rt2 + 2h/α)2 Q 2 2 Recall that (cf., e.g., [14, Vol I p 55]) v 2 ≤ CM ∇v By the above and thanks to Gronwall’s inequality we get v (56) − 2≤e t CM Re1 h v0 +CM h2 Re2 2 T0 + v0 2 2 + (2h Rt2 + 2h/α)2 Q 2 Moreover, t (57) ∇v(s) Re1 ≤ h2 Re1 T0 2 + 2 vz (s) Re2 2 ds + (2h2 Rt2 + 2h/α)2 Q 2 t + h v0 2 + v0 254 CHONGSHENG CAO AND EDRISS S TITI Therefore, by (51), (52), (56) and (57) we have (58) t 2+ v(t) ∇v(s) Re1 ∇T (s) Rt1 t + vz (s) Re2 Tz (s) 2+ Rt2 2 + 2 ds + T (t) 2 2 + α T (z = 0)(s) 2 ds ≤ K1 (t), where (59) K1 (t) = 2(h2 Rt2 + h/α) Q 2 t 2 + h v0 + + CM h2 Re2 + h Re1 t T0 + v0 2 2 + (2h2 Rt2 + 2h/α)2 Q 2 3.2 L6 estimates Taking the inner product of the equation (38) with |v|4 v in L2 (Ω), we get 1d v dt + Re2 6 + Re1 |∇v|2 |v|4 + ∇|v|2 |v|2 |vz |2 |v|4 + ∂z |v|2 |v|2 dxdydz Ω (v · ∇)v − =− dxdydz Ω Ω z −h ∇ · v(x, y, ξ, t)dξ −[(v · ∇)v + (∇ · v) v] + f k × v z T (x, y, ξ, t)dξ − −∇ h −h −h ∂v + (v · ∇)v + (v · ∇)v ∂z z −h T (x, y, ξ, t)dξdz · |v|4 v dxdydz Integrating by parts we get (60) (v · ∇)v − − Ω z −h ∇ · v(x, y, ξ, t)dξ ∂v · |v|4 v dxdydz = ∂z Since f k × v · |v|4 v = 0, (61) then by (36) and the boundary condition (28) we also have (v · ∇)v · |v|4 v dxdydz = (62) Ω Thus, by (60)–(62), 1d v dt + 6 + Re2 Re1 |∇v|2 |v|4 + ∇|v|2 |v|2 |vz |2 |v|4 + ∂z |v|2 |v|2 Ω dxdydz Ω dxdydz 255 PRIMITIVE EQUATIONS =− (v · ∇)v − (v · ∇)v + (∇ · v) v Ω z −∇ −h T (x, y, ξ, t)dξ − z −h h −h · |v|4 v dxdydz T (x, y, ξ, t)dξdz Notice that by integration by parts and boundary condition (28), − (v · ∇)v − [(v · ∇)v + (∇ · v) v] Ω z −∇ −h h T (x, y, ξ, t)dξ − z −h −h · |v|4 v dxdydz T (x, y, ξ, t)dξdz (∇ · v) v · |v|4 v + (v · ∇)(|v|4 v) · v − v k v j ∂xk (|v|4 v j ) = Ω z − −h T (x, y, ξ, t)dξ − z −h h −h T (x, y, ξ, t)dξdz ∇ · (|v|4 v) dxdydz Therefore, by Cauchy-Schwarz inequality and Hălder inequality we obtain o 1d v dt + 6 + Re1 Re2 dxdydz Ω M −h +C |∇v| |v|5 dz −h M |T | +C M |v| ≤C dxdydz Ω |vz |2 |v|4 + ∂z |v|2 |v|2 |v| ≤C |∇v|2 |v|4 + ∇|v|2 |v|2 M −h −h |v| dz −h M |∇v| |v|4 dz −h 1/2 −h |v|6 dz |∇v| |v| dz |∇v| |v| dz −h L4 (M ) |v| dz +C M |∇v| |v| dxdydz −h M 1/4 |v| dz dxdy −h dxdy 1/2 Ω |v| dz 4 1/2 ≤C v 1/2 4 dxdy 1/2 1/2 dxdy dxdy |∇v|2 |v|4 dz |T | +C |∇v| |v|4 dz −h M −h 1/2 0 +C |v|2 dz dxdy −h dxdy 1/4 |v| dz dxdy 256 CHONGSHENG CAO AND EDRISS S TITI 1/2 × Ω |v| dz −h M 1/4 |∇v| |v| dxdydz dxdy 1/2 +C |T | L4 (M ) Ω M 1/4 |∇v| |v| dxdydz |v| dz −h dxdy By using Minkowsky inequality (46), we get −h M 1/2 |v| dz 1/2 ≤C dxdy |v|12 dxdy −h dz M By (45), |v|12 dxdy ≤ C0 |v|6 dxdy M |v|4 |∇v|2 dxdy + M |v|6 dxdy M M Thus, by Cauchy-Schwarz inequality, (63) −h M 1/2 |v|6 dz 1/2 ≤C v dxdy |v|4 |∇v|2 dxdydz L6 (Ω) + v Ω L6 (Ω) Similarly, by (46) and (42), (64) −h M ≤C −h v 1/2 |v| dz ∇v L6 (M ) L2 (M ) + v 1/2 ≤C dxdy |v|8 dxdy −h dz M dz ≤ C v L2 (M ) 6( ∇v + v 2) , and (65) M −h 1/4 |v| dz dxdy ≤C ≤C ≤C 1/4 |v| dxdy −h dz M v −h 3/2 v 3/2 L6 (M ) ∇v ∇v 1/2 1/2 L2 (M ) + v Therefore, by (63)–(65) and (41), 1d v 6 + |∇v|2 |v|4 + ∇|v|2 |v|2 dxdydz dt Re1 Ω |vz |2 |v|4 + ∂z |v|2 |v|2 dxdydz + Re2 Ω 1/2 + v 1/2 L2 (M ) dz 257 PRIMITIVE EQUATIONS 1/2 ≤C v ∇v 1/2 v 3/4 3/2 |∇v|2 |v|4 dxdydz 1/2 +C v Ω ∇v 1/2 v 6 1/2 +C v +C T ∇v 6( |∇v|2 |v|4 dxdydz + v 2) Ω 1/2 ∇T 1/2 v 3/2 ∇v 1/2 + v 1/2 1/2 |∇v|2 |v|4 dxdydz Ω Thanks to the Young and the Cauchy-Schwarz inequalities, d v dt 6 + Re1 |∇v|2 |v|4 + ∇|v|2 |v|2 dxdydz Ω |vz |2 |v|4 + ∂z |v|2 |v|2 dxdydz Re2 Ω ≤ C v ∇v v + C v ∇v + C T ∇T 2 6 2 + 2 +C v 2 v 6 By (58) and Gronwall inequality, we get (66) 6 v(t) t + Re1 |∇v|2 |v|4 dxdydz Ω + Re2 |vz |2 |v|4 dxdydz ≤ K6 (t), Ω where K6 (t) = eK1 (t) (67) v0 H (Ω) + K1 (t) Taking the inner product of the equation (27) with |T |4 T in L2 (Ω), and by (27), we get 1d T dt 6 Rt1 + Rt2 |∇T |2 |T |4 dxdydz + Ω |Tz |2 |T |4 dxdydz + α T (z = 0) Ω 6 Q|T |4 T dxdydz = Ω z v · ∇T − − −h Ω ∂T ∂z ∇ · v(x, y, ξ, t)dξ |T |4 T dxdydz By integration by parts and (36), (68) − v · ∇T − Ω z −h ∇ · v(x, y, ξ, t)dξ ∂T ∂z As a result of the above we conclude 1d T 6 |∇T |2 |T |4 dxdydz + + dt Rt1 Ω Rt2 + α T (z = 0) 6 |T |4 T dxdydz = |Tz |2 |T |4 dxdydz Ω Q|T | T dxdydz ≤ Q = Ω T 258 CHONGSHENG CAO AND EDRISS S TITI By Gronwall, again, (69) T (t) ≤ Q H (Ω) t + T0 H (Ω) 3.3 H estimates 3.3.1 ∇v estimates First, we observe that since v is a strong solution on the interval [0, T∗ ) then Δv ∈ L2 ([0, T∗ ), L2 (M )) Consequently, and by virtue of (36), Δv · n ∈ L2 ([0, T∗ ), H −1/2 (∂M )) (see, e.g., [10], [32]) Moreover, and thanks to (36) and (37), we have Δv · n = on ∂M (see, e.g., [35]) This observation implies also that the Stokes operator in the domain M , subject to the boundary conditions (37), is equal to the −Δ operator As a result of the above and (36) we apply a generalized version of the Stokes theorem (see, e.g., [10], [32]) to conclude: ∇ps (x, y, t) · Δv(x, y, t)dxdy = M By taking the inner product of equation (35) with −Δv in L2 (M ), and applying (36) and the above, we reach d ∇v dt 2 + Δv Re1 2 (v · ∇)v + [(v · ∇)v + (∇ · v) v] · Δv dxdy + = M f k × v · Δv dxdy M Following similar steps as in the proof of 2D Navier-Stokes equations (cf e.g., [10], [32]) one obtains (v · ∇)v · Δv dxdy ≤ C v M 1/2 ∇v Δv 3/2 Applying the Cauchy-Schwarz and Hălder inequalities, we get o (v à )v + (∇ · v) v · Δv dxdy ≤ C M M ≤C −h M ≤C −h M × −h M ≤ C ∇v 1/2 1/2 |v| |∇v| dz −h |v| |∇v| dz 1/4 1/2 |Δv|2 dxdy dxdy M 1/4 |v|4 |∇v|2 dxdydz Ω |Δv| dxdy |∇v| dz |∇v| dz 1/4 2 |v| |∇v| dz |Δv| dxdy 1/2 −h Δv dxdy 259 PRIMITIVE EQUATIONS Thus, by Young’s and Cauchy-Schwarz inequalities, d ∇v dt 2 Δv Re1 + 2≤C ∇v 2 v +C ∇v 2 +C Ω |v|4 |∇v|2 dxdydz + C v By (58), (66) and thanks tothe Gronwall inequality, we obtain ∇v (70) 2 + t Re1 |Δv|2 ds ≤ K2 (t), where K2 (t) = eK1 (t) (71) 3.3.2 vz v0 H (Ω) estimates Since u = vz , it is clear that u satisfies z ∂u + L1 u + (v · ∇)u − ∂t (72) + K1 (t) + K6 (t) −h ∇ · v(x, y, ξ, t)dξ ∂u ∂z + (u · ∇)v − (∇ · v)u + f k × u − ∇T = Taking the inner product of the equation (72) with u in L2 and using the boundary condition (28), we get 1d u dt 2 ∇u Re1 + 2 + ∂z u Re2 z (v · ∇)u − =− −h Ω − 2 ∂u ∂z ∇ · v(x, y, ξ, t)dξ · u dxdydz (u · ∇)v − (∇ · v)u + f k × u − ∇T · u dxdydz Ω From integration by parts we get (73) − z (v · ∇)u − −h Ω ∇ · v(x, y, ξ, t)dξ ∂u ∂z · u dxdydz = Since (f k × u) · u = 0, (74) then by (73) and (74) we have 1d u dt 2 + ∇u Re1 =− 2 + ∂z u Re2 2 ((u · ∇)v − (∇ · v)u − ∇T ) · u dxdydz Ω ≤C (|v|) |u| |∇u| dxdydz + T Ω ∇u 260 CHONGSHENG CAO AND EDRISS S TITI ≤C v u ≤C v ∇u + T ∇u 1/2 3/2 ∇u + T ∇u u By Young’s inequality and Cauchy-Schwarz inequality, we have d u dt 2 + ∇u Re1 2 ∂z u Re2 + 2≤C u ∇v v ≤C 2 +C T + v 2 u 2 +C T 2 By (58), (66), (70), and Gronwall inequality, (75) 2 vz + t Re1 ∇vz (s) 2 Re2 + t vzz (s) 2 ds ≤ Kz (t), where 2/3 Kz (t) = e(K2 (t)+K6 (76) (t))t v0 H (Ω) + K1 (t) 3.3.3 ∇v estimates By taking the inner product of equation (26) with −Δv in L2 (Ω), we reach d ∇v dt 2 + Δv Re1 2 + ∇vz Re2 z (v · ∇)v − =− −h Ω |v| |∇v| + −h Ω ≤C v L6 (Ω) ∇v +C M −h ∂v ∂z ∇ · v(x, y, ξ, t)dξ z +f k × v + ∇ps − ∇ ≤C 2 −h |∇v| dz|vz | + L3 (Ω) |∇v| dz Δv −h · Δv dxdydz T (x, y, ξ, t)dξ −h |∇T | dz |Δv| dxdydz |vz ||Δv| dz dxdy + C ∇T Δv Notice that by applying Proposition 2.2 in [5] with u = v, f = Δv and g = vz , we get M −h |∇v| dz −h |vz ||Δv| dz dxdy ≤ C ∇v 1/2 vz 1/2 ∇vz 1/2 Δv 3/2 As a result and by (43) and (44), we obtain d ∇v dt ≤C 2 + v Δv Re1 L6 (Ω) 2 + + ∇v ∇vz Re2 1/2 vz 1/2 2 ∇v 1/2 Δv 3/2 + h ∇T Δv 261 PRIMITIVE EQUATIONS Thus, by Young’s inequality and Cauchy-Schwarz inequality, d ∇v dt 2 Δv Re1 + 2 ∇vz Re2 + ≤C 2 L6 (Ω) v + ∇v 2 ∇v 2 vz 2 + C ∇T 2 By (58), (66), (70), (75) and thanks to Gronwall inequality, we obtain ∇v (77) 2 t Δv(s) Re1 + 2 ∇vz (s) Re2 + ds ≤ KV (t), 2 where 2/3 KV (t) = eK6 (78) (t) t+K1 (t) Kz (t) H (Ω) v0 + K1 (t) 3.3.4 T H estimates Taking the inner product of equation (27) with −ΔT − Tzz in L2 (Ω), we get ∇T 1d + 2 Rt1 + α ∇T (z = 0) 2 dt 1 ∇Tz ΔT + + Rt1 Rt2 + Tz 2 z v · ∇T − = −h Ω ≤C 2 + α ∇T (z = 0) 2 + Tzz Rt2 2 ∇ · v dξ Tz − Q [ΔT + Tzz ] dxdydz (|v| |∇T | + |Q|) |ΔT + Tzz | dxdydz Ω + −h M ≤C v ∇T +C ∇v + Q ≤C + Q 1/2 Δv 1/2 −h 2 |Tz | |ΔT + Tzz | dz + ∇Tz Tz + ∇Tz 2 ∇T 1/2 ΔT ΔT ΔT v |∇v| dz 2 + ∇v + ∇Tz 1/2 2 Δv + Tzz 1/2 + Tzz 2 ΔT + Tzz 1/2 2 2 dxdy + ∇Tz 2 + Tzz 3/2 2 1/2 1/2 Tz 1/2 1/2 ΔT 2 + ∇Tz 2 + Tzz 3/2 By Young’s inequality and Cauchy-Schwarz inequality, d ∇T 2 + Rt1 ≤C v + α ∇T (z = 0) dt 1 ΔT + + Rt1 Rt2 + Tz 2 + ∇v 2 Δv 2 ∇T 2 ∇Tz 2 2 + α ∇T (z = 0) + Tz 2 By (66), (77), and Gronwall inequality, we get + C Q 2 2 + Tzz Rt2 2 262 (79) CHONGSHENG CAO AND EDRISS S TITI ∇T + 2 + Tz 2 1 + Rt1 Rt2 t ΔT 2 Rt1 ∇Tz + α ∇T (z = 0) Tzz 2 Rt2 + α ∇T (z = 0) 2 + 2 ds ≤ Kt , where (80) 2 Kt = eK6 (t) t+KV (t) T0 H (Ω) + Q 2 Existence and uniqueness of the strong solutions In this section we will use the a priori estimates (58)–(79) to show the global existence and uniqueness, i.e global regularity, of strong solutions to the system (26)–(31) Theorem Let Q ∈ H (Ω), v0 ∈ V1 , T0 ∈ V2 and T > 0, be given Then there exists a unique strong solution (v, T ) of the system (26)–(31) on the interval [0, T ] which depends continuously on the initial data Proof As indicated earlier the short time existence of the strong solution was established in [15] and [33] Let (v, T ) be the strong solution corresponding to the initial data (v0 , T0 ) with maximal interval of existence [0, T∗ ) If we assume that T∗ < ∞ then it is clear that lim sup − t→T∗ v H (Ω) + T H (Ω) = ∞ Otherwise, the solution can be extended beyond the time T∗ However, the above contradicts the a priori estimates (75), (77) and (79) Therefore T∗ = ∞, and the solution (v, T ) exists globally in time Next, we show the continuous dependence on the initial data and the the uniqueness of the strong solutions Let (v1 , T1 ) and (v2 , T2 ) be two strong solutions of the system (26)–(31) with corresponding pressures (ps )1 and (ps )2 , and initial data ((v0 )1 , (T0 )1 ) and ((v0 )2 , (T0 )2 ), respectively Denote by u = v1 − v2 , qs = (ps )1 − (ps )2 and θ = T1 − T2 It is clear that (81) ∂u + L1 u + (v1 · ∇)u + (u · ∇)v2 ∂t z ∂u ∇ · v1 (x, y, ξ, t)dξ − − ∂z −h +f k × u + ∇qs − ∇ (82) z −h ∇ · u(x, y, ξ, t)dξ z −h θ(x, y, ξ, t)dξ = 0, z ∂θ ∇ · v1 (x, y, ξ, t)dξ + L2 θ + v1 · ∇θ + u · ∇T2 − ∂t −h z ∂T2 ∇ · u(x, y, ξ, t)dξ = 0, − ∂z −h ∂θ ∂z ∂v2 ∂z 263 PRIMITIVE EQUATIONS (83) u(x, y, z, t) = (v0 )1 − (v0 )2 , (84) θ(x, y, z, 0) = (T0 )1 − (T0 )2 By taking the inner product of equation (81) with u in L2 (Ω), and equation (82) with θ in L2 (Ω), we get 1d u dt 2 + ∇u Re1 =− 2 + uz Re2 2 z (v1 · ∇)u + (u · ∇)v2 − −h Ω z − − −h ∇ · v1 (x, y, ξ, t)dξ ∂v2 · u dxdydz ∂z ∇ · u(x, y, ξ, t)dξ z f k × u + ∇qs − ∇ −h Ω ∂u ∂z θ(x, y, ξ, t)dξ · u dxdydz, and 1d θ dt 2 + ∇θ Rt1 =− 2 + θz Rt2 2 + α θ(z = 0) z v1 · ∇θ + u · ∇T2 − −h Ω z − −h 2 ∇ · v1 (x, y, ξ, t)dξ ∂T2 ∂z ∇ · u(x, y, ξ, t)dξ ∂θ ∂z θ dxdydz By integration by parts, and the boundary conditions (28) and (29), we get (85) z (v1 · ∇)u − − −h Ω (86) − z v1 · ∇θ − −h Ω ∇ · v1 (x, y, ξ, t)dξ ∇ · v1 (x, y, ξ, t)dξ ∂u · u dxdydz = 0, ∂z ∂θ · θ dxdydz = ∂z Since f k × u · u = 0, (87) and by (85), (86) and (87) we have 1d u dt 2 + ∇u Re1 =− 2 + uz Re2 2 z (u · ∇)v2 · u dxdydz + Ω Ω −h ∇ · u(x, y, ξ, t)dξ ∂v2 · u dxdydz, ∂z then 1d θ dt 2 + ∇θ Rt1 =− 2 + θz Rt2 2 + α θ(z = 0) z (u · ∇)T2 θ dxdydz + Ω Ω −h 2 ∇ · u(x, y, ξ, t)dξ ∂T2 θ dxdydz ∂z 264 CHONGSHENG CAO AND EDRISS S TITI Notice that (u · ∇)v2 · u dxdydz ≤ ∇v2 (88) u u u 1/2 ∇u 3/2 , ∇θ 1/2 ∇u Ω ≤ C ∇v2 (u · ∇)T2 θ dxdydz ≤ ∇v2 (89) 2 θ u ≤ C ∇T2 θ 1/2 Ω Moreover, z Ω −h ∇ · u(x, y, ξ, t)dξ ≤ −h M ≤ −h M dxdy 1/2 0 |∂z v2 | dz −h |∇u| dz −h M |∂z v2 | |u| dz −h 1/2 |u| dz dxdy 2 × −h |∇u| dz −h M |∇u| dz ≤ ∂v2 · u dxdydz ∂z dxdy |∂z v2 |2 dz dxdy −h M |u|2 dz dxdy By Cauchy-Schwarz inequality, (90) −h M 1/2 |∇u| dz ≤ C ∇u dxdy By using Minkowsky inequality (46) and (41), we obtain (91) M 1/2 |u| dz −h ≤C dxdy ≤C 1/2 −h −h |u| dxdy dz M |u||∇u| dz ≤ C u ∇u , and (92) M −h 1/2 |∂z v2 | dz dxdy ≤C −h ≤C 1/2 −h |∂z v2 |4 dxdy dz M |∂z v2 ||∇∂z v2 | dz ≤ C ∂z v2 ∇∂z v2 265 PRIMITIVE EQUATIONS Similarly, z (93) −h Ω ∇ · u(x, y, ξ, t)dξ ∂T2 θ dxdydz ∂z ≤ C ∇u ∂z T2 1/2 ∇∂z T2 1/2 θ 1/2 ∇u 3/2 ∇θ 1/2 Therefore, by estimates (88)–(93), we reach 1d u 2 2 + θ dt + ∇u Re1 2 + uz Re2 + θz Rt2 2 + α θ(z = 0) ≤C ∇v2 + ∂z v2 +C ∇T2 +C ∇u 1/2 2 + ∇θ Rt1 2 2 ∇∂z v2 1/2 1/2 1/2 ∇θ ∇u 2 1/2 1/2 ∇∂z T2 2 ∂z T2 u 1/2 θ θ 1/2 1/2 ∇θ By Young’s inequality, we get d u dt 2 ≤C ∇v2 + ∇T2 + ∂z v2 2 ∇∂z v2 2 + ∂z T2 × 2 ∇∂z T2 + θ 2 ∇∂z v2 (s) 2 u 2 2 Thanks to Gronwall inequality, u(t) 2 + θ(t) × exp C 2 ≤ t 2 u(t = 0) ∇v2 (s) + ∂z T2 (s) 2 + θ(t = 0) + ∇T2 (s) ∇∂z T2 (s) 2 2 + ∂z v2 (s) 2 ds Since (v2 , T2 ) is a strong solution, u(t) 2 + θ(t) ≤ 2 u(t = 0) 2 + θ(t = 0) 2 2 exp{C KV t + Kt t + Kz KV + Kt } The above inequality proves the continuous dependence of the solutions on the initial data; in particular, when u(t = 0) = θ(t = 0) = 0, we have u(t) = θ(t) = 0, for all t ≥ Therefore, the strong solution is unique Acknowledgments We are thankful to the anonymous referee for the useful comments and suggestions This work was supported in part by NSF grants No DMS-0204794 and DMS-0504619, the MAOF Fellowship of the Israeli Council of Higher Education, and by the USA Department of Energy, under contract number W-7405-ENG-36 and ASCR Program in Applied Mathematical Sciences 266 CHONGSHENG CAO AND EDRISS S TITI Florida International University, Miami, FL E-mail address: caoc@fiu.edu Dept of Mathematics and Dept of Mechanical and Aerospace Engineering, University of California, Irvine, CA and Dept of 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Navier-Stokes equations with the free boundary condition, J Appl Math & Optimization 38 (1998), 1–19 (Received March 2, 2005) (Revised November 14, 2005) ...Annals of Mathematics, 166 (2007), 245–267 Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics By Chongsheng Cao and Edriss... the global existence and uniqueness (regularity) of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics Introduction Large. .. and [33] the authors set up the mathematical framework to study the viscous primitive equations for the atmosphere and ocean circulation Moreover, similar to the 3D Navier-Stokes equations, they